We have definitely found that the different SMP hold different challenges for teacher understanding. For example, MP 7 (look for and make use of structure) and MP 8 (look for and express regularity in repeated reasoning) are often challenging at first to make sense of and distinguish.  These two were so challenging for teachers participating in our work that we supplemented the other explorations included in the PD materials by creating a video that provides examples of each. We'll be making the final version of that video available soon. Or in cases like MP 6 (attend to precision), there can be a tendency to make assumptions based on the name alone -- e.g., mistakenly thinking that precision is just about accurate numerical answers, and therefore missing the focus on precise communication. As for MP 3 (Construct viable arguments and critique the reasoning of others) -- you make a good point about the challenge of digging into the nuances of this practice. We have noticed that with both this practice and MP 1 (Make sense of problems and persevere in solving them), there can be a tendency at first for teachers to feel like these are happening all the time (but focusing only at a surface level). So, it requires more work to help push to a greater level of specificity about what we want to see happening when engaging in MP 1 or MP 3 (e.g., as you point out -- reasoning from definitions).

Hi Wendy - in the field test, the materials were mostly used by teachers in same school district, or at the same school, with a few exceptions where the materials were tested in regional PD settings. We did, however, have many mixed-grade groups with middle and high school teachers participating together. We developed two versions of the PD course: one for groups of middle grades teachers, and one for groups of high school teachers; and then we developed a pathway that integrates the two versions for mixed-grades groups. We did put a lot of thought into designing both the problems that teachers engage with in the PD, as well as the structure of the PD sessions themselves, to support teachers in engaging in the mathematics themselves as learners and looking at their own mathematical thinking (as well as learning about their colleagues' approaches to mathematical thinking). Many of the mathematics tasks are low-threshold, high-ceiling problems to accommodate a range of mathematical knowledge. The PD materials do also contain student dialogues, which offer examples of mathematical thinking using three fictitious students; those dialogues are designed to be discussed as interesting exemplars, rather than representations of typical student responses -- within those discussions, we did sometimes hear that teachers considered that thinking not possible with their own students. As for your suggestion about use with principals and other facilitators -- yes, yes, yes! We would love to see a version of the materials for that audience and have put some effort into attracting funding for that work. EDC has a history of developing mathematics PD programs for administrators, through the Lenses on Learning program, and we would love to continue that work.

Hi Libby - thanks for your interest! One source of information about our work is our website, mathpractices.edc.org, which offers many of the materials included in our PD curriculum (though not the PD materials themselves, which we expect to be available from NCTM in the fall). Another source of information is our speaker presentation materials from our 2017 NCSM Annual Meeting session in San Antonio, available at ncsm.org under Session 1106 if you are an NCSM member. We also have two recent articles in NCTM journals about the Standards for Mathematical Practice: Debunking Myths about the SMP by Victor Mateas and What is "Repeated Reasoning in MP8? lead author Paul Goldenberg.

Your question about curriculum materials is a good one and I think there is a lot of potential for curriculum materials to highlight the opportunities to attend to mathematical thinking. One model for an approach to doing this in mathematics is in the new edition of the elementary program Investigations in Number, Data, and Space by TERC, in which the authors have paid careful attention to how to support teachers in thinking about children's thinking in the materials themselves. I'd highly recommend looking at their approach. From their website: "Support material for teachers—an essay about the two practices highlighted and assessed in each unit and sidebars that indicate opportunities for engaging students in all of the practices—provides images of what these practices look like in the elementary classroom, explains how math practices interact with math content, and offers guidance for helping young students learn how to use these practices in their mathematical work."

I think there is also some wonderful work going on in thinking about formative assessment in mathematics and several of my colleagues at EDC have written about this for classroom teachers, see for example Uncovering Student Thinking about Mathematics in the Common Core by Tobey and Fagan and Routines for Reasoning by Kelemanik, Lucenta, and Creighton. While not examples of curriculum per se, these books provide classroom resources and are strong examples of materials that help teachers think about supporting, soliciting, and responding to student thinking, and may be helpful models to consider.

Hi Miriam - in our twenty-hour PD course, we have a few different areas of focus. Specifically, the goals of the course are to:

• Increase awareness of the SMP.
• Develop understanding of the SMP and their connection to the Standards for Mathematical Content.
• Cultivate capacity to identify the SMP in student thinking.
• Develop ability to plan how to use instructional tasks in order to encourage student engagement in the SMP.

In a subset of the PD sessions, we have teachers focus explicitly on planning at the unit and lesson level, incorporating a focus on mathematical thinking and the SMP. We also address instructional practices, including orchestrating discussions, that support the development of mathematical practice. Our focus is on helping teachers build their own understanding of the SMP and then to reflect upon their own practice and plan for modifications.

Our research on the impact of the PD course, however, was largely limited to the effects the course had on teachers awareness and understanding of the SMP and the extent to which teachers felt prepared to teach with attention to the SMP. We did not have the resources to look closely at what was happening in classrooms using for example an observational measure. Our evaluation partners at TERC did interview a subset of teachers across participating sites, many of whom did report that the PD experience had an impact on their teaching, largely in how teachers planned and enacted curriculum with students to highlight opportunities to engage in mathematical practice and enhance discussion to focus on thinking.